Is the inclusion $\partial M\hookrightarrow M$ homotopy equivalent to the inclusion of a subcomplex into a CW-complex, i.e. is there a CW-complex $X$ with a subcomplex $Y$ and homotopy equivalences $g\colon X\rightarrow M$ and $h\colon Y\rightarrow \partial M$, such that the obvious diagram commutes up to homotopy?

I think this must be true, but I can't find a reference. I would be also interested, if someone has one for the smooth case.

2 Answers
2

I have no idea at the moment where to find a reference for the specific result you seek. However, it can be deduced from the following fact: topological manifolds (paracompact and Hausdorff) are absolute neighbourhood retracts, and thus have the homotopy type of CW-complexes. This is briefly stated in corollary 1 of Milnor's article On spaces having the homotopy type of a CW-complex (published in Trans. Amer. Math. Soc. 90, 1959, pages 272-280). An extensive discussion of these matters is given in the thesis A topological manifold is homotopy equivalent to some CW-complex by Aasa Feragen. For the specific case of compact manifolds, an elementary proof is given in the appendix of Hatcher's book Algebraic topology (see corollary A.12 there).

We can now prove the result you state. Let $Y$ be a CW-complex admitting a homotopy equivalence $f : Y \to \partial M$ to the topological manifold $\partial M$. Denote by $j : \partial M \to M$ the inclusion of the boundary of $M$, and factor the composition
$$ Y \overset{f}{\longrightarrow} \partial M \overset{j}{\longrightarrow} M $$
as the inclusion of a sub-CW-complex $i : Y \to X$ followed by a weak equivalence $h : X \to M$. The weak equivalence $h$ is a homotopy equivalence since both its domain and its target have the homotopy type of CW-complexes.

Finally, let me briefly justify why we can take $i : Y \to X$ to be the inclusion of a subcomplex of a CW-complex. The usual construction of the factorization above produces $(X,Y)$ as a relative CW-complex, i.e. $X$ is obtained from $Y$ by adding cells in increasing order of their dimension. Now observe that this construction can be modified to inductively build $X$ as a CW-complex itself: at each stage of the construction of $X$, we can replace the attaching maps of the cells with homotopic cellular maps using the cellular approximation theorem.

Can't you assume that $i: Y \rightarrow X$ is an inclusion of a subcomplex by replacing $X$, if needed, by the mapping cylinder of $i$? Anyways, great answer!
–
Piotr PstrągowskiJan 31 '14 at 10:40

Dear @Piotr Pstrągowski, I am afraid I do not see why the mapping cylinder of $i$ must be a CW-complex. After all, cells of lower dimension in $X$ may attach to cells of higher dimension in $Y$. The real issue is guaranteeing $X$ is a CW-complex in the end.
–
Ricardo AndradeJan 31 '14 at 10:44

Oh, you're right. I believe we could first deform $i$ to be cellular, then take the mapping cyllinder, then use the homotopy extension property to make the resulting diagram commutative and not just commutative up to homotopy. But that's not really any easier than the argument in your answer.
–
Piotr PstrągowskiJan 31 '14 at 11:01

Kirby and Siebenmann proved that any compact topological manifold has the homotopy type of a finite complex (announced in Kirby, R. C.; Siebenmann, L. C. On the triangulation of manifolds and the Hauptvermutung. Bull. Amer. Math. Soc. 75 1969 742–749, with proofs in their book "Foundational essays on Topological Manifolds").

I just want to make a couple of tangential remarks about this result of Kirby and Siebenmann. First, their result is much finer and harder than the statement that manifolds have the homotopy types of CW-complexes. A celebrated generalization of this result is Borsuk's conjecture that compact ANRs have the homotopy type of finite CW-complexes, which was finally proved by James West and announced in Compact ANR's have finite type (Bull. Amer. Math. Soc. Volume 81, Number 1, 1975, pages 163-165).
–
Ricardo AndradeJan 31 '14 at 13:54